The relative accuracy of quadrature formulae of the Cote's closed type

TECHNISCHE UNiVgnglTEIT DELFT

LUCHTVAART- i'd mi Anï.C»^^A"LJ^ 12

Kluyverweg 1 - 2629 HS D E L ^ '

^^^-12 Juli 1950

C R A N F I E L D

The R e l a t i v e Accuracy'' of Q u a d r a t u r e Formulae of t h e C o t e s ' C l o s e d Type

Bv

-S. Kirkby, Ph.D., A.F.R.Ae.-S., of the Department of Aerodynamics,

oOo

SUMARY

-Quadrat'ore'formulae, such as those discovered by Gregory, Nev/ton, Simpson and Cotes, which are derivable by

integration of Lagrange's interpolation formula between definite limits, are classified as CoLes' Type Formulae. Yftien the functional values at the end-points of the range

of inte^ation are used the corresponding formulae are said to be of the 'closed type\

It is shovm that, for closed type formulae, the error awe to application of a 2n-strip forraula is in general less than that due to a (2n + 1)-strip forroula over the same range of integration when using the same tatular i-.terval of th? ':!rg\Jinent,

1. IMTRODUCTIOH,

It iz v."cll knovm that in numerical integration by closed type formulae, Simpson's one-third rule (2--3ti'ip) is mcro accijrate than Simpson's three-eighths rule (3-strip) when the tabvu.ar

interval is the sam.e. The relative accuracy, based on the leading term in the error series, is k/9 provided that the integrand and its fourth derivative are continuous throughout and at the limits of the range of integration.

Similarly, and genera].ly, for closed type formulae, it is shown that a 2n-strip formula is more accurate thar a (2n + 1)-strip formula when the same tabular interval is used. The relative accuracy, again based on the leading error term and assuming continuity of the integrand and its (2n + 3)'tb derivative, is in fact alvTays less tha.n 5/4, however large the value of n,

The above comparisons inherently necessitate using an even number, 2n(2n + 1 ) , of strips for the integration. This is not alv/ays convenient, particularly when the integration is to be evaluated from au even number of experimental points, yielding an odd number of strips. In these circumstances it may be expedient to use an odd-strip formula over part of the range of integration, 2, RELATIVE ERROR OF CLOSED TYPE FORivIUI.AE.

In generaJ. we may express a Cotes' closed type quadrature formula in the form

•N p i f(x) dx = ', / (x) dx + Rjj

N

2 Jp f(r) + n^ (1)

r=1

where jè{x) is a polynomial of degree N - 1 , which coincides with f(x) for N values of the argument, and the multipliers Jj. are

Christoffel numbers whose numerical values depenc' upon N and are given by the equation

N N - 1 •

N-r ƒ M l ' I dv

y - r The i-tinainder, R^j , after N terms is discussed below.

We distinguish between the cases N odd or even by writing N = 2n + 1 or N = 2n + 2 respectively. Then for closed type fonrralae (Hef.l)

2 n + 3 (2n+ 3 ) , ^ . , .

and

2n + 2 ' •* ' 2n + 2

R„ . = -h^''"^5f(2n+3) ( | ) c . , ...(3)

where 1 < f <C 11 and h is the tabular interval of the argument,

The coefficients

G^^ ,. ,

Cp p ^ ^ expressible in terras of

Bernoulli's polynomials of order (2n + 3) by the equalities

(2n+ 3)^ ^ (2n+

^)

Bo. . . (2) + B ^ , (1)

2n + 3 2n + 3

'^2n + 1

"^^'

and

(2n + 3)

2B (2)

C^ „ = . (5)

2n + 2

(2n -r 3)1

Since

(n) (n) (n-'l)

B ( x + 1) ^ E' (x) + V B (x) , (6)

it follows that, after putting x = 1 and 0 in equation (6) and

substituting in equations (4) and (5), the coefficients C ,

Cp p m.a7 also be expressed in terms of generalised Bernoulli's

numbers by the equations

(2n + 3) (2;i+ 2) (2n + i)

2B 3B B ^

2n + 3 2n + 2 2n + 1

Cp . =

:

+ + .,.(7)

^"•^ ^ (?n +

3)1

(2n+ 2)1 (2n + 1),'

and

2n + 3 2n + 2

^2n+-2 "

•"

• ^Q)

'-^^ ^

(2n+ 3)J (2n-. 2 ) !

These last two equations represent a convenient form for evaluating

the Teading term in the error series,

The generalised Bernoulli's polynomials and numbers

employed above ivere discovered by Norlund (Ref.3) and ere also

described by Milne-Thornson (Ref,4). The generalised Bernoulli's

polynojTiials of crv..er n are given by

^"^^ =ySi B^") (x) ,

(et_|)n ^ V :

'

-

^

while the generalised Bernoulli's numbers of order n are given by

t" _ 'Cr t-__ ^(n)

For the purposes of comparison and to determine the relative accuracy of two quadrature formulae of the closed type, it is essential that we use both the saune range of integration

and the same tabular interval h in each case, ¥e shall therefore select

h = , " - ^ , • (9) (2n)(2n + 1)

The relative accuracy of the numerical integiations resulting from (2n + 1) applications of a 2n-strip closed type formula and frcm 2n applications of a (2n + 1)-strip closed type formula is then E2n ^ 2 n + 1 = ( 2 n . 1 ) E ^ ^ ^ , ^ =2n . 2 t ^(10) , ( 1 1 ) by equations (2) and (3).

Prom equations (11), (4) and (5) E 2n

'--) f::>^^^'i:'M

E„ , 2n 2B^^"

*

^hl)

2n 2

since B 2n + "5 (2) is negative when 2 -^ 2n + 4, i.e. when

n > 1.

Hence

^2n

^ J_

(12)

^2n + 1 ^ when n 5» 1•

Therefore, as proved in equation (12), the error due Lo (2n -f 1) applications of a 2n-strip formula of the closed type is less than three-quarters of the error due to 2n applications of a (2n + 1)-strip formula of the closed type when applied over a given range of integration using the same tabular interval in eaah c '.se,

It is vrarthy of note that the error due to either of the above formulae will be decreased, although the relative accuracy will remain unchanged, if the tabular r'nterval giv ;n by equation (9) ia subdivided into an integral number, p , of suo-intervals of o-xtent

N - 1

form^ila

p(2n)(2n + 1)

From equations (11), (7) and (8) v/e obtain an alternaoive

^ =

^ . U l

J ^ ' 2 : ? - 3 ( 2 n . 3 ) B ( f j ^ ) . ( 2 n . 3 ) ( 2 n . 2 V ^ - i ) )

^2m1 2" 1 fflt^-'+J)., 2(2n+3)E(fi' J

^ 2n+3 2n+2

= (13)

which is suitable for computing particular values of the relative accu'.'acy, Ep /Ep . , from a table of generalised Bernoulli's numbers such as given by Milne-Thomson (Ref.l).

If in equation (13) we substitute n = 1, l^ogether vd'^h the values

B^3) _ _ i . ^(4) ^ 251_ 3(5) ^ . 4 7 5 3 • 4 ' 4 3 5 ~T2

the relative accuracy of three applications of Simpson's one-third rule as against two applications of Simpson's three-eighths rule is found to be

E^ 4 E , 9

Di.inc--n refers to this result in a recent Note in which he investigates the errors due to the use of certain quadrature formulae (Rc:f,4).

Similarly, when y/e substitute n = 2, 3 in equation (13),

vf'j

obtain

and

E

^5

"6 .

128

275

3888

8183

0.465

0,475 .

From the above general results, it is advisof^. that Co'res' closed tj'pe quadrature formulre using an even number of strips (odd number of ordinates) should whenever convenient be employed in preference to the corresponding odd-strip formulae with one more strip (or ordinate),

Milne-Thomson, L.M, The Calculus of finite Differences. London, MacMillau (1933).

Milne-Thomson, L.M. Two Classes of Generalised P^lynomia's. Proc. London Math. Soc.,(2), 35

(1933)

Norlund, N.E. Memoire sur les polynomes de Ber.'noulli. Acta. Math. 43 {'32?), pp. 15^ - 196, Duncan, W. J, Niomerical Evaluation of Ordinary and

Singular Defini'^^e Integrals.

Note on Computational Methods,!'o.5. The College of Aeronautics (^^l^''),